Factoring polynomials with 2 variables

Question

How do I factor polynomials when they have 2 variables like these:

30x²-+11xy-30y²

and

45x²-59xy+6y²

There are so many possibilities to try. Is there any other easier way of doing this besides trying every possibility?

Answer 1

Factoring 30x²+11xy-30y² is not much different than factoring 30x²+11x-30.  For

30x²+11x-30, here's a non-traditional approach that might help.  Form the equation  30x²+11x-30=0 , and use the quadratic formula (perhaps skip ahead a few sections in your text book to find it).  Get x = -6/5 and x = 5/6.  Use these numbers to help factor the original polynomial.  Start by using the opposite numbers we found x to equal.  (x+6/5)(x-5/6).  FOILing this, get    x²+(11/30)x - 1.  Clearly not what we were looking for, but by multiplying by 30 we get the original polynomial.

30(x+6/5)(x-5/6) = 30x²+11x-30.  The 30 can be distributed (carefully) into the two binomials, using a factor of 5 in the first and a factor of 6 in the second.  This gets us (5x+6)(6x-5).  Similarly, the answer to 30x²+11xy-30y² is (5x+6y)(6x-5y).

I'm not sure if this method is easier than trial & error, but it is a bit more concrete.

Answer 2

Please show me step by step how to factor

this problem. Thank you

 

a7 b6-a4b3+a3b3